https://file.aiquickdraw.com/mathbot/file/1729490335669ussfy1pg.jpg

Let's solve the given problems step by step using the provided functions:

We have:
- \( f(x) = 2x^2 \)
- \( g(x) = 3x + 1 \)

### 1. \( f(x) + g(x) \)
\[
f(x) + g(x) = 2x^2 + (3x + 1) = 2x^2 + 3x + 1
\]

### 2. \( f(x) - g(x) \)
\[
f(x) - g(x) = 2x^2 - (3x + 1) = 2x^2 - 3x - 1
\]

### 3. \( f(g(x)) \)

Substitute \( g(x) = 3x + 1 \) into \( f(x) = 2x^2 \):
\[
f(g(x)) = 2(3x + 1)^2 = 2[(3x + 1)(3x + 1)]
\]
Expanding the square:
\[
(3x + 1)(3x + 1) = 9x^2 + 6x + 1
\]
So:
\[
f(g(x)) = 2(9x^2 + 6x + 1) = 18x^2 + 12x + 2
\]

### 4. \( f(x)^2 \)

\[
f(x)^2 = (2x^2)^2 = 4x^4
\]

### 5. \( f(x)^3 - g(x) + g(x)^2 \)

First, calculate \( f(x)^3 \), \( g(x) \), and \( g(x)^2 \):
- \( f(x)^3 = (2x^2)^3 = 8x^6 \)
- \( g(x) = 3x + 1 \)
- \( g(x)^2 = (3x + 1)^2 = 9x^2 + 6x + 1 \)

Now, combine them:
\[
f(x)^3 - g(x) + g(x)^2 = 8x^6 - (3x + 1) + (9x^2 + 6x + 1)
\]
Simplifying:
\[
= 8x^6 - 3x - 1 + 9x^2 + 6x + 1 = 8x^6 + 9x^2 + 3x
\]

---

Let me know if you want further clarification or details!

### Relative Questions:
1. What is the general process for finding \( f(g(x)) \) for different types of functions?
2. How would the result of \( f(x)^2 \) change if \( f(x) \) was a linear function instead of quadratic?
3. Can you describe a scenario where the combination \( f(x) - g(x) \) represents a real-world problem?
4. How do we expand binomials like \( (3x + 1)^2 \) efficiently?
5. What are the general rules for combining functions like \( f(x) \) and \( g(x) \)?

### Tip:
Always simplify functions step by step, especially when combining terms or applying operations like squaring or cubing.

Given the function f(x) = 2x^2 and g(x) = 3x + 1, find: 1. f(x) + g(x) 2. f(x) - g(x) 3. f(g(x)) 4. f(x)^2 5. f(x)^3 - g(x) + g(x)^2

Learn how to solve function operations such as addition, subtraction, composition, and powers with the functions f(x) = 2x^2 and g(x) = 3x + 1. Detailed solutions for Grades 9-11 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation